The Rayleigh–Jeans law agrees with experimental results at large wavelengths (low frequencies) but strongly disagrees at short wavelengths (high frequencies). This inconsistency between observations and the predictions of classical physics is commonly known as the ultraviolet catastrophe.[1][2] Its resolution in 1900 with the derivation by Max Planck of Planck's law, which gives the correct radiation at all frequencies, was a foundational aspect of the development of quantum mechanics in the early 20th century.

In 1900, the British physicist Lord Rayleigh derived the λ−4 dependence of the Rayleigh–Jeans law based on classical physical arguments and empirical facts.[3] A more complete derivation, which included the proportionality constant, was presented by Rayleigh and Sir James Jeans in 1905. The Rayleigh–Jeans law revealed an important error in physics theory of the time. The law predicted an energy output that diverges towards infinity as wavelength approaches zero (as frequency tends to infinity). Measurements of the spectral emission of actual black bodies revealed that the emission agreed with the Rayleigh–Jeans law at low frequencies but diverged at high frequencies; reaching a maximum and then falling with frequency, so the total energy emitted is finite.

where h is the Planck constant and kB the Boltzmann constant. The Planck's law does not suffer from an ultraviolet catastrophe, and agrees well with the experimental data, but its full significance (which ultimately led to quantum theory) was only appreciated several years later. Since,

even after substituting the value λ=c/ν{\displaystyle \lambda =c/\nu }, because Bλ(T){\displaystyle B_{\lambda }(T)} has units of energy emitted per unit time per unit area of emitting surface, per unit solid angle, per unit wavelength, whereas Bν(T){\displaystyle B_{\nu }(T)} has units of energy emitted per unit time per unit area of emitting surface, per unit solid angle, per unit frequency. To be consistent, we must use the equality

Depending on the application, the Planck function can be expressed in 3 different forms. The first involves energy emitted per unit time per unit area of emitting surface, per unit solid angle, per spectral unit. In this form, the Planck function and associated Rayleigh–Jeans limits are given by

Alternatively, Planck's law can be written as an expression I(ν,T)=πBν(T){\displaystyle I(\nu ,T)=\pi B_{\nu }(T)} for emitted power integrated over all solid angles. In this form, the Planck function and associated Rayleigh–Jeans limits are given by

In other cases, Planck's law is written as u(ν,T)=4πcBν(T){\displaystyle u(\nu ,T)={\frac {4\pi }{c}}B_{\nu }(T)} for energy per unit volume (energy density). In this form, the Planck function and associated Rayleigh–Jeans limits are given by